It is not critical for register assignment -- in fact, SSA makes register assignment more difficult (see the swap problem; the lost copy problem).
Lifetime analysis is important for register assignment, and SSA can make lifetime analysis easier, but plenty of non-SSA compilers (lower-tier JIT compilers often do not use SSA because SSA is heavyweight) are able to register allocate just fine without it.
Lifetime analysis is important for register assignment, and SSA can make lifetime analysis easier, but plenty of non-SSA compilers (lower-tier JIT compilers often do not use SSA because SSA is heavyweight) are able to register allocate just fine without it.