Does "living in squalor" necessarily imply poverty?

Some definitions of squalor and its adjectival form squalid:

Merriam-Webster

squalor: the quality or state of being squalid
squalid: marked by filthiness and degradation from neglect or poverty

Collins

squalor: the condition or quality of being squalid; disgusting dirt and filth
squalid: dirty and repulsive, esp as a result of neglect or poverty

Wiktionary

squalor: Squalidness; foulness; filthiness; squalidity.
squalid: Extremely dirty and unpleasant

Compact Oxford Dictionary

squalor: the state of being extremely dirty and unpleasant, especially as a result of poverty or neglect
squalid: (of a place) extremely dirty and unpleasant, especially as a result of poverty or neglect

(Ok, now that y’all are suffering from the same word glare/semantic satiation as I am …)

If someone is described as living in squalor, does that automatically imply that they’re doing so because they’re too poor to do otherwise? I know that context can override the usual connotation of words, so for example if I’ve established that someone is a reclusive millionaire, the fact that she’s living in squalor would not necessarily be a contradiction. But if I encounter this phrase on its own, am I wrong to assume poverty?

Yes, you are wrong to assume poverty, but I suspect it's what many of us might do.

By the definitions that you provided above, and my own interpretation, "Living in squalor" does not directly imply poverty. It could just as easily be neglect. For example, if someone of moderate means is a hoarder, they could easily be living in squalor.