Answer:
1) "Equivocating was my only option, the woman was giving me no choice. I didn't want to tell her anything."
2) "The man hid his eyes, he avoided any questions the captain threw at him about his heritage, or why he was there. When it came to decision making, the boy watched him equivocate with kind mannerisms and little nods of his head whenever appropriate. They knew nothing about him, he could've been anyone."
Explanation:
Equivocated isn't actually a word, equivocating is though.
It means, from a quick search online definition,
"to use ambiguous language so as to conceal the truth or avoid committing oneself." Essentially, to dance around a topic by using fancy language, talking without actually saying anything.
I hope all this helps some. :)