Does Nash Equilibrium predict the existence of vaccine reluctance?
It depends on how you build the game. You may have games with several Nash equilibria, with a dominant strategy equilibrium (which is a special case of Nash equilibrium) or no equilibria at all.
Let us consider a game with 3 players, where herd immunity is achieved at 40% and where players incur a small cost for taking the vaccination and make a big gain when herd immunity is achieved. In this case you have 3 Nash equilibria (when 2 players decide to get the vaccine, nobody has an incentive to deviate).
In another game you may have 3 players who don't care about herd immunity, but that want to become immune (say the disease can come back at any time). In this case you have dominant strategy equilibrium: everybody gets the vaccine.
Finally let's construct a game with 2 players, where herd immunity is achieved at 40%. In this scenario players incur a small cost for taking the vaccination, and a bigger cost if they are the only player who takes the vaccination [1]. Again, herd immunity gives a gain. This can be modelled with a payout matrix such as:





PlayerA Vax
PlayerA No Vax




Player B Vax
1,1
2,-1


Player B No Vax
-1,2
0,0




then the game doesn't have a Nash equilibrium (you may have noticed that this is the same payout matrix of the prisoner's dilemma).
To answer your question, a game that includes vaccine reluctance may have several equilibria or none at all. Vaccine reluctance is an a-priori to your model, which can be factored in when building the payout matrix. Then you can analyze the game an try to predict possible outcomes.
[1] For instance if players are members of an anti-vax community/group, they may incur a reputation cost if other members don't get the vaccine.