Can -que be used with negation?

Question

If I want to negate two things, can I combine them with -que? For example, if I want to say "I don't have a cat or a dog", can I say non habeo felem canemque? Or should it rather be non habeo felem canemve?

My intuition is to say non habeo felem nec canem rather than non habeo felem et canem, but I am less sure about the enclitic.

This question arose from TKR's comment to my answer to a recent question by tony. The case there was a little more convoluted, but I want to start with this simple setting and then ask follow-up questions if needed.

Answer 1

non habeo felem et canem would mean "I do not have both a cat and a dog." Because -que generally behaves like et, I assume it would be interpreted identically.

Generally, when negating two elements, Latin will use a parallel structure. I would expect either:

Habeo nec/neque felem nec/neque canem

or

Non habeo (aut) felem aut canem. (Using an enclitic would not change the meaning.)

I would not say non habeo felem nec canem, because nec should be used either as an element of a pair or at the start of a new clause (combining the et that would serve as the clausal conjunction with the non of the second clause).