Does it have to do with unpaired electrons? I’m really unsure.
The answer lies in the formation of the molecular orbitals for H2. When the two s-orbitals of the H atoms combine (overlap), there are two possible combinations – essentially they either add together, or they subtract. The additive combination is called a (sigma) bonding orbital, while the subtractive combination is called an antibonding orbital. Putting electrons into a bonding orbital helps to hold the atoms together (stabilizes the bond), whereas putting them into an antibonding orbital tends to drive them apart (destabilizes the bond). H2^+ has one electron shared between the two H nuclei, and it goes into the lowest energy orbital available… the bonding orbital. That helps to hold the H’s together. In H2, you have two electrons, and both can go into the bonding orbital (with opposite spins). Now you have two electrons helping to bind the atoms together. That’s better than only one electron in the bonding orbital, so H2 is more stable than H2+. When you get to H2^-, you now have three electrons. The first two go into the bonding orbital, but the third won’t fit (Pauli exclusion principle… only 2 electrons can fit into one orbital), so it has to go into the antibonding orbital, which tends to weaken the bond. Roughly speaking, the antibonding electron cancels out the good effect of one of the bonding electrons, so you have approximately the same bonding strength as in H2^+. summing up, H2 is the most stable.
Hope that helps!