A negation of a statement sounds wrong? I have the statement "Any non-negative number that is smaller than any positive number is zero", which I quantified as (assuming $U=\mathbb{R}$)
$$(\forall \: x \ge 0)[(x < \varepsilon)(\forall \: \varepsilon > 0) \implies (x = 0)]$$
Negating:
$$(\exists \: x \ge 0)[(x < \varepsilon)(\forall \: \varepsilon > 0) \land (x \neq  0)]$$
Which I interpret as "There exists a non-negative, non-zero number that is smaller than any positive number". Is this the proper negation? Can a quantifier+predicate statement and its negation have the same or opposite truth value?
 A: 
I have the statement "Any non-negative number that is smaller than any positive number is zero", which I quantified as (assuming $U=\mathbb{R}$)
$$(\forall \: x \ge 0)[(x < \varepsilon)(\forall \: \varepsilon > 0) \implies (x = 0)]$$

When formalising a statement, the convention is that each predicate's quantifiers precede the predicate, and the quantifiers then read from left to right (changing their order generally alters the statement's meaning).
So, correction: $$\forall x{\ge}0\;\Big(\forall \varepsilon{>}0\; x < \varepsilon \implies x = 0\Big).$$

Negating:
$$(\exists \: x \ge 0)[(x < \varepsilon)(\forall \: \varepsilon > 0) \land (x \neq  0)]$$

Similar correction: $$\exists x{\ge}0\;\Big(\forall \varepsilon{>}0\; x < \varepsilon \;\land\; x \ne 0\Big).$$
This means that there exists some positive number such that it is smaller than every positive number.

Can a quantifier+predicate statement and its negation have the same or opposite truth value?

Negating a statement flips its truth value. In the above example, negating a true statement creates a false statement.
