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Oh no they do recognize all the Popes after Pius X including Pope John Paul II, they just named their organization after Pius X because he is a Saint (and probably also because he wrote an encyclical against Modernism, I believe) like the Benedictines named their order after Saint Benedict. Their position is just that they don't believe they are required to listen to the Pope because they think they are doing the right thing by only saying the Tridentine mass and following the practices of the Pre-Vatican II church. They even accept Vatican II, with some hesitation, just understand that the SSPX aren't sedvecantists, who say that there was no real Pope ordained after Pope Pius XII's death in 1958 (because then they would have to accept the teachings of the Second Vatican Council). I believe the SSPX only went into schism in the late 1980's (1988 or '89) when Lefebvre ordained four Bishops to succed him, which the Pope said he could not do, because he was dying and wanted Bishops to carry on the work of his order by ordaining Priests.

I found the date. It was June 30, 1988 that the Society of Saint Pius X went in schism. I don't even think there was an official proclomation of schism, because they automatically encured it by Archbishop Lefebvre ordaining four Bishops. Shortly after that Pope JP II issued Ecclesia Dei which allowed the Latin mass to be said with an indult from a local Bishop. As a result of Ecclesia Dei organizations like the Priestly Fraternity of Saint Peter (FSSP) which say only the latin mass and are not in schism, were formed.

This is a picture of Archbishop Marcel Lefebvre ordaining four Bishops which caused his group, the SSPX, to become schismatic.