The purpose of the questions in Dread is to build a picture of the character which will indirectly answer the larger questions of what keeps the character going when things get tough. As host, you should first make sure players understand this larger premise, and buy into it. Each answer should ideally provide some piece of the puzzle, some insight into what will keep a character motivated when a situation turns grim. Especially important to consider is this bit from the section "Filling Out a Questionnaire" on page 21 of the core rules:
When filling out a character questionnaire, you should always assume
the presence of a silent “and why?” at the end of each question. It
will create a better understanding of the character. The answer will
cover more ground, and there will be less room for misunderstanding
during the game itself.
If a player rejects the premise of every question, you might have to have a more serious discussion. But if he's only done this on one question, you have three options.
The first and easiest option is to just let it slide. The rules suggest not every question needs an answer, and if the player's other answers have given you what you need, just move on. If the player has considered it and this is his answer, move on. If the player has a block and can't answer this one, move on.
The key here is to move on. Keep things going.
The second option is one I would not recommend, but I include it because it is an option. Remind the player of the reason for the questions, and insist the premise be honored. Each questionnaire is unique, and the mere presence of the question is a declaration. Yes, your character did have a pet, and yes, that pet died. It's right there on your character sheet. Again, I don't recommend this option, but it's still an option.
The last option is my favorite. Ask the player to honor the spirit of the question while allowing him to reject the premise. This is where that rule I quoted comes in. Push for the "why." "You didn't have a pet? Why?" This option is my favorite because it permits the player a degree of control while still pushing for details that will help build a larger picture.
You mention the concept of "Yes, and..." It's a great concept. But remember, you can also make use of this concept. You can also say "Yes, and..." to any answer you think could have a bit more detail. As host, it's your job to help the players build their characters. "Yes, and..." is a powerful tool for helping them.
Hope this helps.