What is the most professional name for "squiggly bracket"?

I think curly brackets is the most professional name. See Wikipedia for more details.

curly braces (or 'braces' for short)

This is more common than curly brackets, to design blocks in programming languages.

See Why curly braces? (WebArchive) for more.

Curly braces are one means of denoting a lexical scope. Lexical scopes are blocks of code from which names do not escape

See also:

• history and use of parentheses in programming languages, from the beginning of programming to the present day.
• wordiq definition.

I always thought { was officially a brace.

[ is a bracket

( is a parenthesis

With "bracket" also referring to any one of the three.