Air traffic controller reported assigning two crossing altitudes to an air carrier aircraft and the flight crew continued through both altitudes after they had acknowledged both restrictions.

Narrative

Had a sequence of 3 aircraft going into JAC with one aircraft to the west at 130 on a heading for spacing. I specifically remember clearing Aircraft X to cross HOMVA (IAF for RNAV X RWY 01) AOA (at or above) 140 because I did not have track control of the 130 aircraft yet. They read it back correctly. This of course did not matter because Aircraft X kept descending below 140; getting as low as 120 in a 129 MIA (Minimum IFR Altitude) box. When I saw them descending through 135 I re-cleared them to cross HOMVA AOA 130 (still good with MIA/approach plate); which they read back correctly yet they kept descending. When I saw they were not stopping at 130 as cleared I issued a low altitude alert and instructed them to climb to 130. They read this back and complied with the crossing restriction. Later they explained that the aircraft system was programmed for the lowest crossing altitude on the approach plate; which is 117 at HOMVA.I don't think there is any reason why the approach plate crossing altitudes should be that low around the Tetons. There shouldn't be a way the plane's FMS or whatever can take these guys down to 117 in such a mountainous area. The main issue was that Aircraft X just hit whatever default 'cleared approach' button in their computer and didn't put in the crossing altitude I gave them; so maybe the air carrier should look at their systems and implement something that forces them to enter a crossing altitude.