Linguistic name for verbs like 'want', 'expect', 'beg',

Within theoretical linguistics, these are called Exceptional Case Marking verbs.

http://en.wikipedia.org/wiki/ECM_verb

"You" is actually the subject of "to come." In most cases, infinitives can't have subjects, but verbs like "beg" and "expect" allow their infinitive complements to have subjects.

As a side note, I don't believe that "eat" is an ECM verb, but if it was, what would "We eat you to come" mean? You should choose your sentences carefully so that they could mean something, because otherwise you might accidentally think something is impossible just because it doesn't make sense. For example:

• I will eat [the tomato growing in the yard].
• *I will eat [the tomato to grow in the yard].

vs.

• I expect [the tomato to grow in the yard].

Edit: There is another class of verbs which look sort of like ECM verbs, but have a slightly different structure. These are called Object Control verbs.

• We persuaded you [PRO to come].

http://www.ling.upenn.edu/~beatrice/syntax-textbook/ch9.html#oc