The other answers already provided some intuition so I will try to be slightly more technical (although not so that a non-economist would not be able to follow).
tl;dr:
Government cannot prevent passing tax burden just by controlling prices (at best it can mitigate it and even that at the expense of workers). Generally to prevent the passing of the burden it would have to outright nationalize the firms and institute centrally planed economy. Centrally planed economies are generally not endorsed by economists as they lead to inefficiency and lower standards of material welfare.
This does not mean we should have no taxes, neither that taxation and redistribution cannot improve welfare of the poor (see part III of this answer), but it does mean that generally speaking tax burden in market economy will be distributed widely throughout the economy whether we like it or not.
Full Answer:
Distribution of Tax Burden in Market
In order to answer this question let us start by discussing how the tax burden gets shifted around.
First, actually companies themselves can't decide how much burden get passed down to the consumers. Hence the whole premise of the original statement:
One of the arguments against taxing large businesses is that they'll just pass the costs on to their customers
Is actually myth, but as most myths it has a kernel of truth. In actuality the way how tax burden is distributed across economy depends on parameters of demand and supply and firms or even government has very little to none control of those parameters. For example, a general rule of thumb when it comes distribution of tax burden between consumers and producer is that the side of the market (supply or demand) that is more elastic bears the lower share of tax burden (see Mankiw Principles of Economics (8th ed.) - section 6-2 Taxes (starting on pp 121).
In order for consumer's to bear no tax incidence their demand would have to be either completely elastic and supply cannot be completely inelastic. Government can't do much about this given that demand is mainly given by consumer preferences. Preferences could be changed through propaganda for example but I can't even begin to fathom the sort of indoctrination that would be required to turn inelastic demand, for let's say insulin, into completely elastic demand (that would be feat that would made even big brother full of envy). Or government would have to prevent both price and quantity to change in any way which is tantamount to nationalizing the firms.
Practical Stylized Example
This can be illustrated by a stylized example. Suppose demand is $Q_d=100-p$ and supply is $Q_s= 10+p$, where $p$ and $Q$ are price and quantity respectively. Furthermore, let us assume simple production function where product is directly proportional to employment $Q=L$. We have to start with an example with no tax as the tax burden is calculated as the difference in producer and consumer surplus vis-a-vis the situation with no tax. In this stylized example without tax in an equilibrium the situation will be as follows:
$$Q_d = Q_S \implies 100-p= 10+p \implies p^* = 45€ \implies Q^* = 55 $$
where stars indicate equilibrium. The consumer surplus (benefit consumers derive from buying goods on the market) is always given as the area under the demand curve and above equilibrium price:
$$CS= \int_{0}^{Q^*} p(Q_d) dQ - p^*Q^*$$
where $p(Q_d)$ is an inverse demand function since by convention price goes on $y$-axis. By evaluating the integral we find that in this case $CS=1512.5€$.
The producer surplus $PS$ (analogue of $CS$ for producers is given by the area below equilibrium price and above supply curve so:
$$PS= p^*Q^*- \int_0^{Q^*} p(Q_s) dQ,$$
where $p(Q_s)$ is the inverse supply curve. In this case $PS=962.5€$
Now if government imposes some tax $t=\\\$1$ per product sold solely on producers we get some pass of tax burden but not full. This would change supply to: $Q_s=10+p-t \implies Q_s = 9+p$.
$$Q_d = Q_S \implies 100-p= 9+p \implies p^* = 45.5€ \implies Q^* = 54.5 $$
So we can see that even though the tax was set to $1€$ and firms in this example are rational profit maximizes - hence they want to to pass as much tax to consumers as possible - they only could pass $\\\$0.5$ on consumers not the whole $1€$. Furthermore, the new $CS_{\text{ after $t$}} = 1485.125€$ and hence tax burden on consumers is: $CS-CS_{\text{ after $t$}}= 1512.5-1485.125= 27.375€$.
New Producer surplus is $940.125€$ so their burden is $PS- PS_{\text{ after $t$}} = 962.5 - 940.125 = 22.375€$. So both producers and consumers share the tax burden. We also see workers got 'screwed' since now the employment is lower as based on our previous assumptions on production $Q^*=L^*=54,5$ so employment falls by $0.5$. Generally speaking only in special cases where one side of the market is completely elastic and the other isn't we would get a situation where only one side of the market bears the full burden of a tax and the fortunes of workers are intimately tied to the fortunes of producers.
This example showcases important fact: it is the market as a whole that decides how tax burden is distributed - companies don't decide this themselves. Government also can't decide this just by stating who is to pay the tax de jure as in this case government de jure levied tax solely on firms.
Can Price Controls Eliminate the Pass of Tax Burden?
The answer is generally no, although it can change the distribution of the burden. Let us continue the previous example and let us suppose that government in addition to $t=1€$ fixes prices at the old equilibrium $p=\bar{p}=45€$. What will happen now? Now we will have:
$$Q_d = 100 - \bar{p} =55$$
$$Q_s = 10 + \bar{p} -t = 9+45 = 54$$
So despite demand being $55$ only $54$ units of product will be produced by firms (i.e. the $Q$ we use in calculations of $CS$ and $PS$ is $54$). This gives us $CS=1512€$ and $PS=486€$. So even though the burden on consumers now will only be $0.5€$ which is lower, it cannot be completely eliminated with exception of situations when there would not be any burden to begin with (also generally results would not be so impressive in this case the price happened to fall on a portion of demand curve where the elasticity is relatively large). Furthermore, note employment now falls by whole $1$ which is twice as much as before when we had tax without any price controls. Lastly note that in order to reduce the burden on consumer by meagerly $26.875€$ we had to raise the burden on producer by whooping $476.5€$. That is as long as you think consumers are not at least approx. $18 \times$ more deserving this would be bad trade (and bear in mind not all producers are high income individuals).
This is not even whole story. The above example is trivial, oversimplified static case. In real life equilibrium price changes constantly and government has no way of identifying it. Government can't simply keep the price at original $45€$ if peoples preferences or any other relevant parameters of demand and supply change. Even with big data and modern IT technology government has no direct way of objectively observing these changes because many of them (like preferences) are purely in the heads of consumers who do not have incentive to provide accurate information to government (and even if they would have they might not realize their preferences until they actually make choice). The further the true equilibrium price strays away from the government imposed price the worse off people will be (assuming there are no special cases of market failures and even then crude price setting would be terrible idea) so this is bound to cause problems. I won't explain this point further because this answer is already too long but you can see detailed explanation in any undergraduate textbook (for example ibid. Principles of Economics (8th ed.) ch 6-1 controls on prices pp. 112). I also think that the fact that this is considered to be such a point of consensus as to be included in virtually any undergraduate textbook speaks for itself, if you want to know more consider posting it as a separate question.
Furthermore, in order for government to eliminate the burdens on consumers completely it would have to set not only price but also quantity produced and by extension the employment. At this point even if we would retain de jure private ownership firm is de facto nationalized as government would do all price setting, quantity setting, employment setting choices. Given that the profits are actually rewards firms get for optimizing their $p$ and $Q$ there is no need at this point for private ownership. If government would set $p$ and $Q$ in a way that firms are not profitable enough for keeping the business alive to be the best choice for the owner, the owner will close down business, and if $p$ and $Q$ are high enough why should private individual get rewards when government did all work for that person?
Conclusion
As the example above illustrates in order for the government to prevent tax incidence to be passed to consumers, while also keeping employment constant, the government must not control just price but also quantity produced. At that point we turned our economy into centrally planned economy. There are several countries that historically tried to centrally plan their economy, the only surviving case (that I know of) is North Korea.
Economists generally don't believe having centrally planned economy is good way of organizing economy. This is due to the so called problem of economic calculation (or some authors call it the problem of socialist calculation) which states that in the absence of market prices & private ownership it is impossible to perform economic calculations in order to optimize production. Again I won't be dissecting this issue here now as books have been written just on this subject alone (although I encourage you to post it as a separate question if interested).
Lastly, none of the above is argument against taxation. Government is needed for optimal provision of public goods and also (depending on peoples' political preferences) redistribution (see Public Choice III by Dennis Mueller for detailed overview of various reasons why we need government). The point is simply that it is not possible for government to choose freely who will and who won't bear the burden of taxation.
PS: Of course, the example above is extremely stylized. I wanted to keep everything on undergraduate level and non-economist friendly even though I included technical example - because in my experience it is easier for people to understand concept when they are presented with some case they can go through (even if the case is simplistic).
If you want to explore this issue further then I recommend as further reading:
- Joseph E. Stiglitz. Economics of the Public Sector. - whole text
- Mankiw. Principles of Economics - whole text
- MWG. Microeconomic Theory. - selected chapter discussing welfare and taxation
- Varian. Microeconomic Analysis - selected chapter discussing welfare and taxation
