Is this with the out-of-the-box setup?
Unless you play extremely lightly with your right hand, very low strings are going to be a problem – even on fretted, but especially on fretless. The board slap/buzz is actually a big part of the tone and the appeal of fretless IMO. I don't want it, then I play fretted.
This is not to say that the neck is up to snuff. Maybe is isn't. But no instrument comes out of the box perfectly set up for every player.
FWIW, I learned on fretless, and played it exclusively for the first 20 years of my bass playing.
My instinct and experience says that you should be going higher with your strings, not lower. That will give you more of the upright feel and tone. And if you want to stay low, then embrace the board slap and learn to use it...or start playing with a feather-light right hand.
While I agree that higher action will give you a more upright-y tone, I have to respectfully disagree about embracing the fretline noise. The sizzle from protruding frets is a very different sound than the mwaah you get from low action, in my experience. It's a very non-musical hashy sound, like distortion or the sizzle of a noisy amp, and seems to be worse with rounds than flats. On my ebonol neck when the fretlines stood proud, the issue was not very dependent on string height, unless the strings were WAY off the board.
If the neck surface is good, low action = more mwah, with none of the sizzle/buzz that you get from surface imperfections like proud fretlines. The sizzle/buzz from proud fretlines is noise and nothing worse embracing.
FWIW, I played exclusively fretless through the 90's and into the '00s, and currently play fretless about 1/3 of the time. Have owned both lined and unlined fretlesses from Squier to Rob Allen. Not trying to pick a fight, but if you haven't experienced proud fretlines, you may not realize how different they are from ordinary mwaah.
Doc, did the Squier in question have rounds or flats on it? My experience has been that rounds are less forgiving of surface imperfections on the fretboard, but curious what you've found.