Two-Weapon Fighting requires light, one-handed, melee weapons, which the stones are not
[...] When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. [...]
Nothing states that the stones are considered light or that they are melee weapons, in fact, they are used to make spell attacks not weapon attacks. Thus, they cannot be used with Two-Weapon Fighting. Even if you got the Dual Wielder feat they would still not work:
[...] You can use two-weapon fighting even when the one handed melee weapons you are wielding aren’t light. [...]
This still requires that you use one-handed melee weapons, which the stones are not (nor is the sling you could use with the stones).
It's unclear whether magic stone uses the Attack action
The spell might require its own unique action to hurl the stones and not the Attack action. I believe this to be the case as it makes a spell attack and does not explicitly mention the Attack action like other features do; for example, the Sun Soul Monk's Radiant Sun Bolt:
You gain a new attack option that you can use with the Attack action. This special attack is a ranged spell attack with a range of 30 feet [...]
Because of this, the spell cannot benefit from Extra Attack as it requires the Attack action:
[...] You can attack twice, instead of once, whenever you take the Attack action on your turn. [...]
Either way, nothing is going to break
Any GM is perfectly allowed to rule either way and have the stones use the Attack action or not, though that would not change the fact that the stones are not light, melee weapons and cannot be used for TWF. I can't see any problems arising from allowing them to work with Extra Attack, so it would be a fine interpretation/ruling to me.
Personally, I would say magic stone requires its own action, but that's just me. I could easily see it being ruled or interpreted the other way around.