I don't know how you got that idea of it being humiliating judging from the text. The squire is actively looking for lords or knights to serve, although he comments something about serving ecclesiastical lords as not being a very good job for they are not that affluent. The thing he actually finds humiliating is serving certain lords that will eventually turn you into a servant for everything ("que os habéis de convertir en malilla").

Being a squire to a higher up would be one of the jobs for the lowliest of the hidalgos, and that would be a stretch in the case of the squire protagonist of the third chapter. If you read carefully, you'll notice that he doesn't give away where he is from, he just uses very vague terms: " were not the houses there, but where I was born, sixteen leagues from there, in the Costanilla of Valladolid". This one, in the original phrasing, is even more ambiguous as one cannot really tell whether he was born in the Costanilla of Valladolid, or 16 leagues from there. The Costanilla was a rather well known street in Valladolid (today, Bajada de la Libertad), where a lot of jews and conversos lived, so the squire's hidalguía may beentirely ficticious.

That would be further supported by the ways of greeting that the squire had (besoos las manos and other stupidities). That specific part of the Lazarillo de Tormes is a paraphrasis of fray Antonio de Guevara's letter to Alonso de Mendoza, Bishop of Palencia, in which Guevara strongly condemns all of those greetings, especially "besoos las manos", as improper of a good courtier or a good christian.

So, the squire is out of a job, but trying to get one, while still trying to keep up the pretense of his hidalguía. Manual labour would be beneath an hidalgo, and there would also be certain jobs that required a rather long apprenticeship (carpenters, typesetters, metallurgists, bookbinders, butchers), which would exclude him from getting them. A classic way for hidalgos to rise on the social hierarchy would be to become jurists, getting employed by a court, the Inquisition (our squire is not getting this one either as they performed background checks), or be lawyers. This would, of course, require to go to university. If the squire didn't, it may be because he couldn't read, or because he did not know Latin, a language necessary for jurists.

So, the poor squire did not have much of a choice if he wanted to keep up the pretense of his hidalguía while still trying to find someone to serve