word to describe a person who catches the sexual overtone in a normal conversation

It's an idiom, not a word, but we sometimes say one's mind is in the gutter.

EDIT: As for specific words, one could say, "You have a dirty mind!" My computer's on-board thesaurus offered these synonyms for that sense of the word dirty:

indecent, obscene, rude, naughty, vulgar, smutty, coarse, crude, filthy, bawdy, suggestive, ribald, racy, salacious, risqué, offensive, off-color

Of those, I think suggestive, vulgar, smutty, risqué, and off-color perhaps fit most closely with what you're getting at.

As an aside, ELU had an earlier discussion about the word naughty.

Dirty-minded:

obsessed by sex and tending to see the sexual or obscene potentialities in things

From popular culture, we'd normally refer to that person as 'Finbar Saunders'.

you are noticing the ambiguity of a double entendre

I think the word that you might be looking for is innuendo

Hmmm... Just thinking about terms used on me...

Deviant when someone is trying to be nice.

Devious if they think I am trying to make someone uncomfortable on purpose. Nothing more fun than to have a little fun with someone too serious.

Pervert if the person is annoyed at you. Note on this: If a member of the opposite sex calls you a pervert then continues to talk to you, you are in for a good night.

An older person that doesn't know what to think might say twisted.

The deeply religious might say immoral or depraved.

And of course if I have had too much to drink and the conversation is going no where... I am the degenerate.

degenerate -

1. having lost the physical, mental, or moral qualities considered normal and desirable; showing evidence of decline.

Although I doubt many would think of doggy style as a catchy name for a dog grooming business, your husband and his (male) friends who immediately interpreted "it" as being the sexual act performed between two consenting adults, do indeed have dirty minds. Not that it is exclusive to men, but I prefer the expression to have a one-track mind for men and women alike.

The Free Dictionary defines it as: to think entirely or almost entirely about one subject. Longman is less demure and says: to be continuously thinking about one particular thing, especially sex

I believe the word you're looking for is:

licentious
adjective formal disapproving
US /laɪˈsen.ʃəs/ UK /laɪˈsen.ʃəs/

(especially of a person or their behavior) sexual in an uncontrolled and socially unacceptable way

licentious. 2020. In dictionary.cambridge.com
Retrieved on 25 July 2020 from https://dictionary.cambridge.org/us/dictionary/english/licentious

Also see:
lascivious or lewd

However, IMO, your husband and his friends are an accurate representation of the reaction most people would have upon seeing a sign for a dog grooming business called "Doggie Style". Especially in an urban area people will probably think the double entendre was intentional and likely drive more customers your way. I'm from the Boston area, perhaps that explains my opinion or are the very least gives it some context. Have you seen the movie "Ted"?