If I were you, just let your BF have a DNA. If it is negative, then you could ask her to stop communicating with you or else you will be construed to file a case against her. But if it is positive, then your BF is only oblige to support the CHILD not the mother. Or else, your BF maybe charged of Economic Abuse.
As for now, you can only tell the "girl" that you will only communicate with her once she could afford DNA Testing or else you might be construed to file Extortion (im not sure if this is the case you can file against her. Pls ask an atty).
But here's the definition of extortion in the Phil
18-3-207. Criminal extortion� aggravated extortion.
(1) A person commits criminal extortion if:
(a) The person, without legal authority and with the intent to induce another person against that other person's will to perform an act or to refrain from performing a lawful act, makes a substantial threat to confine or restrain, cause economic hardship or bodily injury to, or damage the property or reputation of, the threatened person or another person; and
(b) The person threatens to cause the results described in paragraph (a) of this subsection (1) by:
(I) Performing or causing an unlawful act to be performed; or
(II) Invoking action by a third party, including but not limited to, the state or any of its political subdivisions, whose interests are not substantially related to the interests pursued by the person making the threat.
(1.5) A person commits criminal extortion if the person, with the intent to induce another person against that other person's will to give the person money or another item of value, threatens to report to law enforcement officials the immigration status of the threatened person or another person.
(2) A person commits aggravated criminal extortion if, in addition to the acts described in subsection (1) of this section, the person threatens to cause the results described in paragraph (a) of subsection (1) of this section by means of chemical, biological, or harmful radioactive agents, weapons, or poison.
(3) For the purposes of this section, "substantial threat" means a threat that is reasonably likely to induce a belief that the threat will be carried out and is one that threatens that significant confinement, restraint, injury, or damage will occur.
(4) Criminal extortion, as described in subsections (1) and (1.5) of this section, is a class 4 felony. Aggravated criminal extortion, as described in subsection (2) of this section, is a class 3 felony.
'Blackmail' would fall under the crime classification of EXTORTION and would be punished according the penalty for that particular crime in whatever state it is you live in.