I wouldn't stay with any man who delighted in calling me a name he knew I hated. That is not a lover, it's a bully. I would tell any man who tried to bully me in this way that. If I called my husband a name and he said he disliked it I would never call him it again. If I love him why would I? I agree squishy does make you think of fat and loose squashable flesh. If my husband called me "squishy" I'd take it as a sign to exercise more and would get angry with him.
this has been taken all out of context though squishy as a name (without the nemo connotation it IS not nice) but because it IS linked, its a nice thing (if you like icky pet names)
he said something like you shall be my squishy and my squishy you shall be or something like that, its supposed to be a cute thing, sounds like you just dont like THAT particular pet name, but choosing one you do like would be a bit fake, pet names are like nick names, they just "happen"