No, you do not have to prove your citizenship, because the payment is not only for US citizens. It is for all US citizens or "resident aliens" for tax purposes, who have a Social Security Number, make less than the income threshold (it phases out at an AGI of between $75,000 and $99,000 for single filers and $150,000 and $198,000 for joint filers), and cannot be claimed by someone else as a dependent on their taxes.
The recovery rebates are provided in the CARES Act (text here), section 2201. The parts that limit eligibility for the rebate are the following:
The income limitation:
(c) Limitation based on adjusted gross income.—The amount of the
credit allowed by subsection (a) (determined without regard to this
subsection and subsection (e)) shall be reduced (but not below zero)
by 5 percent of so much of the taxpayer’s adjusted gross income as
(1) $150,000 in the case of a joint return,
(2) $112,500 in the case of a head of household, and
(3) $75,000 in the case of a taxpayer not described in paragraph (1)
Disallowing it to nonresident aliens, people who can be claimed as dependents, and estates and trusts:
(d) Eligible individual.—For purposes of this section, the term
‘eligible individual’ means any individual other than—
(1) any nonresident alien individual,
(2) any individual with respect to whom a deduction under section 151
is allowable to another taxpayer for a taxable year beginning in the
calendar year in which the individual’s taxable year begins, and
(3) an estate or trust.
And disallowing it to people who don't have a Social Security Number (or whose spouses don't have a Social Security Number if filing jointly):
(g) Identification number requirement.—
(1) IN GENERAL.—No credit shall be allowed under subsection (a) to an
eligible individual who does not include on the return of tax for the
(A) such individual’s valid identification number,
(B) in the case of a joint return, the valid identification number of
such individual’s spouse, and
(C) in the case of any qualifying child taken into account under
subsection (a)(2), the valid identification number of such qualifying
(2) VALID IDENTIFICATION NUMBER.—
(A) IN GENERAL.—For purposes of paragraph (1), the term ‘valid
identification number’ means a social security number (as such term is
defined in section 24(h)(7)).
(B) ADOPTION TAXPAYER IDENTIFICATION NUMBER.—For purposes of paragraph
(1)(C), in the case of a qualifying child who is adopted or placed for
adoption, the term ‘valid identification number’ shall include the
adoption taxpayer identification number of such child.
(3) SPECIAL RULE FOR MEMBERS OF THE ARMED FORCES.—Paragraph (1)(B)
shall not apply in the case where at least 1 spouse was a member of
the Armed Forces of the United States at any time during the taxable
year and at least 1 spouse satisfies paragraph (1)(A).
(4) MATHEMATICAL OR CLERICAL ERROR AUTHORITY.—Any omission of a
correct valid identification number required under this subsection
shall be treated as a mathematical or clerical error for purposes of
applying section 6213(g)(2) to such omission.
To answer your question, someone on a work visa will almost certainly qualify for the payment if their income is below the threshold, because they are "resident aliens" for tax purposes since they pass the Substantial Presence Test (except maybe if they arrived this year), and they have Social Security Numbers.