The evidence can only be suppressed if defendant's fourth amendment rights are violated by the police or other law enforcement agencies (evidence obtained as a result of illegal search or seizure) and it is the evidence that the prosecution has against the defendant. There is no remedy afforded to the prosecution as far as suppressing the evidence that the defense has.
Based on the facts of your situation, the prosecution here would be asking for the exclusion of defense evidence from a jury trial based on the gag violation through in limine motion (motion to excluded the evidence) and the defense attorney would be dealing with two issues here - validity of the gag order (usually issued as prohibition to discuss the case with the media) and defendant's right to a fair trial. It all depends on what evidence was obtained and how important that evidence for the defense, and whether that evidence is material or crucial to the defense.
Apparently the defense attorney didn't act in the best interests of his/her client; it would be better to petition the higher court for the removal of the gag order and then obtain the evidence legally rather than illegally. An appellate counsel, if the case goes as far as appeal, may consider arguing a 6th amendment right to an effective counsel claim.
There is not suppression of the evidence available for the prosecution. Suppression is the remedy when the police or government obtains evidence in violation of the 4th amendment. It only applies to government actors.
On a side note - you are presuming that the recording is illegal - whether suppression is a remedy or not. You are also presuming the gag order is valid. Gag orders almost always are directed at the lawyers not to speak about the case to the media. The gag order needs to weighed against the 1st amendment rights of the parties. Gag orders are supposed to make sure there is fairness in the proceedings.
The above information does not establish an attorney client relationship nor is it meant to provide legal advice.
The defense does have disclosure obligations under the discovery statutes. If the evidence is going to be used at trial, the defense may have to disclose it prior to trial or the prosecution may be able to have it "excluded" from the trial, or some other less onerous remedy. Exclusion is not the same as suppression, as pointed out, but the defense is not entitled to avoid statutory procedures and sandbag the prosecutor (unless you have a very schrewd attorney and the right situation!)
I agree with my colleagues.
But a direct answer to your question is that the defendant's attorney would really have to convince the Judge that the defendant's due process would be violated by not letting these recordings in. Give both the gag order and illegal recordings (I'm assuming they were taken without notifying the recorded witness), that would be difficult, and if this case were lost at trial, defense would have to appeal that issue. Part of the appeal would be on the gag order validity grounds my colleague raises.
1) Defendant should talk to their attorney right away about the issues I raise, and those of my colleagues.
2) If everything is as you say, defendant may be facing charges for using this evidence, and needs to weigh that against the value of the evidence. Defendant could be facing charges for the recording alone, regardless of whether it were used.
3) While it would not be "suppression", the same thing IS at stake: yes, one remedy for prosecution would be for the Judge to refuse to admit the evidence.
4) HOW DID THIS HAPPEN? Where was the defendant's attorney? Is the defendant acting alone: what do they expect from their attorney if they don't tell them they're going to do something like this?
The defense has to comply with the discovery rules the same way the DA does with one notable exception: the defense is not under any duty to disclose whether or not the defendant will or will not testify. That is a game day decision!