Actually, it's not exactly correct.
"两个都不要" means"I don't want these two."
literally means" Two两个 ,both都, not不, want要"
But the question is "Do you like....."
In Chinese, it should be "你喜欢......吗?"
你喜欢苹果还是橙子?
The perfect answer should be "我都不喜欢。"
Literally means "I both not like"
Or short version "都不喜欢"
Tips: when you answer somebody's question, you should pay attention to the verb he uses.
If he uses "要.....吗?"(Do you want....?)
You can answer "都不要" as the meaning of "neither"
If he uses"喜欢......吗?"
You should answer"都不喜欢"as the meaning of "neither"
To make it simple,
If you want to use "neither", you use "都不" + verb(that you heard from the question)