Cant was the criminal argot, the language of the beggars and thieves in Merry England, though sometimes it was also used to mean hypocritical or wheedling speech of any kind. The term is first used in English at the end of the fifteenth or beginning of the sixteenth century.
Most dictionaries say that the word comes from the Latin cantare, to sing, though some trace it to a pair of puritanical preaching brothers called Cant. It has also been suggested (many times) that cant comes from the Irish or Scots Gaelic cain(n)t, meaning speech. This suggestion is not new and was not invented by Cassidy. For example, it is given already in R. McCutcheon’s Modern Language Notes in 1921.
Personally, I find the Irish/Gaelic explanation unlikely because it is found in England at such an early time. There is little evidence for linguistic crossover between the two languages as early as this, though there are occasional – and well-attested – words which did enter English at an early period. For example, brat is a Celtic word for a cloak which was already found in Old English (this means a rag in English dialect and may be the origin of brat as in badly-behaved child, though there is another candidate), while the word cros, an Irish version of Latin crux, was brought to the York area by Vikings who had settled in Ireland and gradually displaced the English term rood, so that cross is the standard term for a cross in English today. However, the few real examples are quite conspicuous and it is very unlikely that cant is a word of Irish origin.