This paper is concerned with the analysis of blow-ups for two McKean-Vlasov equations involving hitting times. Let $(B(t); \, t \ge 0)$ be standard Brownian motion, and $τ:= \inf\{t \ge 0: X(t) \le 0\}$ be the hitting time to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) - α\mathbb{P}(τ\le t)$. We provide a simple condition on $α$ and the distribution of $X(0)$ such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time $t \ge 0$. Our approach relies on a connection between the McKean-Vlasov equation and the supercooled Stefan problem, as well as several comparison principles. The second equation is $X(t) = X(0) + βt + B(t) + α\log \mathbb{P}(τ> t)$, whose Fokker-Planck equation is non-local. We prove that for $β> 0$ sufficiently large and $α$ no greater than a sufficiently small positive constant, there is no blow-up and the McKean-Vlasov dynamics is well-defined for all time $t \ge 0$. The argument is based on a new transform, which removes the non-local term, followed by a relative entropy analysis.