Hmm ... if you mean HA migration, yes Proxmox VE supports this; if you mean resetting the machine by a push on the reset button or a power outage (really hard shutdown), then no. This is no migration, this is failover. The VM will restart on the other side (the same as "current" LXC migration).
The case of a "critical HA" (as you wrote), so a fault-tolerant VM, this is currently being implemented on upstream QEMU/KVM and will be included into PVE, yet it is not finally released yet. This enables a kind of a "shadow" of a VM on another machine so if the one fails hard (reset, fire, meltdown...) the other will just jump in (with a small maybe negligible loss). IMHO this is already available in VMware, yet it is very hard to do correctly and almost impossible to do very fast, yet "critical HA" is normally not very fast as all other commercial products show.
The case of a "critical HA" (as you wrote), so a fault-tolerant VM, this is currently being implemented on upstream QEMU/KVM and will be included into PVE, yet it is not finally released yet. This enables a kind of a "shadow" of a VM on another machine so if the one fails hard (reset, fire, meltdown...) the other will just jump in (with a small maybe negligible loss). IMHO this is already available in VMware, yet it is very hard to do correctly and almost impossible to do very fast, yet "critical HA" is normally not very fast as all other commercial products show.