My body my choice (False equivalence?)

It is just a bumper stick slogan. If you have a right to make a choice over your body in one case why would that not apply in the other?

Actually, I think it is a fair "comeback." The problem is that "My body my choice" is a meant to be a sound byte and not an argument. There are too many exceptions to "My body my choice" that make it false/undesirable. For example, I can't strangle someone with my body. The anti-vaxxers are probably doing one of two things:

1) pointing out the problem with the "My body my choice" mantra.
2) being hypocritical and actually adopting the "My body my choice" manta - repurposing it for their own benefit.

Now, there is an argument to be made for a false equivalence if we expand the "My body my choice" mantra to a better argument. For example, it is "my choice" because I am not hurting anyone else and anti-vaxxers are causing massive harm to society in general. Granted, this would be the argument I would suspect one would make, and someone with a pro-life stance would argue that choosing to abort is causing massive harm (certain death to a human). So depending on your moral views abortion, this may or may not be fallacious.

It's not an argument as Dr Bo points out. This is extremely common in political discourse, actually - simplistic one-liners, or clichés, being used as if they were arguments. It's designed ultimately to dismiss a point of view without really reasoning through it properly.

We can parse an implicit argument, though:

P1) If it's my body, it is my choice whether or not to take the COVID vaccine.

P2) It is my body.

C) Therefore I don't have to take the COVID vaccine.

This is a valid argument, but doesn't justify not taking it - it only sets out guidelines for whether one can be compelled to take it or not. Furthermore, most anti-vaxxers modify C) to be 'therefore I shouldn't take the COVID vaccine' (which is a non sequitur; it does not follow that because something is not forced on you that you should decline it).

As Dr Bo also mentions, one could argue a false equivalence between vaccine mandates and abortion bans. But in order to start doing all of those lovely things, we must first get an actual argument out of the person.