Libertarianism entails a specific set of core principles, but it is not a single fixed interpretation of those principles - just like with any other political movement or ideology. An individual self-identified Libertarian's approach to the problem would depend on specific facts available to that individual, and on the individual's understanding of those facts.
I do not count myself as a Libertarian (at most, very tangentially adjacent), but I have spoken with enough people who do that I can imagine a few reactions. Rather than propose a specific solution, I want to lay out the factors that I would expect Libertarians to consider when weighing their options.
Coordination problems
To the extent that government regulation is tolerated under Libertarianism, the purpose of such regulation is to solve coordination problems. There are two classic examples:
The tragedy of the commons: each producer of goods would prefer to exploit a natural resource maximally in order to maximize profit and avoid being out-competed. However, if everyone does this, that resource is depleted in a way that causes lasting harm to everyone.
The boycott: each consumer (or at least, many consumers) of a product wish to discipline a producer for some ethical failing (never mind how deontological or consequential the underlying ethics may be), by switching to a competitor. However, a large number of them would have to agree (and commit) in order for the producer to be meaningfully impacted, whereas each individual would have to pay the competitor a significantly higher price for ostensibly the same good.
Both of these can be viewed as a sort of multi-player Prisoner's Dilemma, which neatly enough explains why a free market fails to solve the problems automatically. Individuals optimizing for personal outcomes steer away from the optimal global equilibrium.
However, this weakness is also a strength: since it is difficult to coordinate, it is also difficult for producers to collude, and thus eliminate market competition (which would naturally raise prices to the point that buyers can bear, as now their only other options are to go without or produce the good themselves). Therefore, a Libertarian requires a compelling case to be made that the coordination problem is a real problem in a specific situation.
Forms of market interference
Taking the object example of eggs from chickens raised with vs. without antibiotics. A Libertarian might take the failure of boycotts as evidence that people do not see a real problem - "if you aren't willing to spend $X more for eggs, how much do you really care?".
On the flip side, in the "tragedy of the commons" framing, we might try to convince Libertarians that antibiotic use in egg farming is an externality: that is, using antibiotics constitutes harm to "the commons", in the form of antibiotic-resistant bacteria. The natural form of regulation to deal with this is a Pigouvian tax. Since measuring an individual farm's production of antibiotic-resistant bacteria, their relative level of resistance etc. would be infeasible, naturally such a tax would be on the antibiotics themselves.
However, before such a measure could be implemented, there would have to be satisfactory evidence of the scale of the problem. While the problem could be broken down in a variety of ways, probably the bottom line would be - how common is it for antibiotic-resistant bacteria from eggs to result in sickness/injury/death to the human population? How strong is the evidence that this problem would become bigger over time, and by how much? Do these bacteria spread away from the farm - that is, does their mere existence increase the chance that salmonella contamination elsewhere in the world will be from an antibiotic-resistant strain?
And what other mitigations are possible? Perhaps there are smarter dosing regimens, for example. Or perhaps there are more cost-effective ways to protect the hens from salmonella infection. And then, a Libertarian might well propose to do away with the American regulation to wash eggs in the production chain. Which brings us to...
Technology
Markets don't only exist to discover prices, but to lower them (in real terms) over time - equivalently, to enable people to purchase more and better things year over year. That is essentially what economic growth consists of, and the reason why capitalism's advocates praise it so highly: because of the goods we have today that we didn't in yesteryear.
So, rather than raising prices on the commons-destroying option, it would be preferable to make the commons-preserving option more economically competitive. It should be noted that a Pigouvian subsidy doesn't really count here; the money for it would have to come from somewhere - i.e., from taxes on something else, which require a separate justification. Aside from that, a subsidy is harder to argue for in a Libertarian framework because it comes across more strongly as favouritism from an already-distrusted government. A Libertarian would need strong evidence that such a measure would actually address a meaningful externality, and not also cause its own problems. (Indeed, I have heard many argue that the corn subsidy in the US results in rampant misuse of the crop.)
An optimistic Libertarian might therefore expect the problem to solve itself, by way of opportunists looking to reduce the cost of antibiotic-free eggs through technology. That could include developing things like
- cheaper vaccinations against salmonella for hens
- cheaper detection mechanisms for salmonella-contaminated eggs
- better treatments to protect eggs (perhaps something like irradiation rather than washing, to avoid breaking down the natural membrane)
A really cheap, reliable salmonella detector could also be marketed to the consumer, as a way to detect spoilage (some people are happy to let eggs sit around in the fridge for weeks, after all).
Finally, I'd like to argue that lab-grown meat, if it became inexpensive and desirable enough, actually would mitigate the issue. This is because consumers would shift their preferences towards lab-grown meat and away from other sources of animal protein (while overall consumer demand is often thought of as insatiable, this doesn't apply so much to food, as we have limited stomach capacity and suffer negative health effects from obesity). This would lower demand for eggs etc., either reducing competitive pressure to produce more or eventually even making them no longer a commercially viable product.
