This is an example of Present Subjunctive.
You use it in a that-clause(but you can omit the that) to express an action or idea is necessary. It takes a bare infinitival(the
plain form of) verb.
Verbs like recommend and suggest are usually used to indicate this form.
This contruction is generally more common in American English than British English. In British English, using should is preferable(like the first sentence)
The last sentence is correct but you don't emphasize the neccessity of eating more vegetables.