Without a recording it's hard to tell, but a native speaker pronounces this like "shouldn't've." You will sometimes see it actually spelled that way in print. The "h" in "have" is silent, the "a" is a schwa (ə), and the "v" is almost unvoiced (because it's hard to switch quickly from the unvoiced "t.") And it's very soft.
A lot of times the _'ve_ or "of" is spoken like a schwa. In quoted dialog, it is sometimes written out as just "a:" "I shouldn't a done it." Other times, because of the "v" being almost unvoiced, it is written as "of," "I shouldn't of eaten it."
So "shouldn't have eaten" becomes "shouldn't've eaten" becomes "shouldn't-a eaten" but the vowel sounds of the "a" and the "e" of "eaten" join and blend.
If you listen carefully there is probably a _faint trace_ of something in there between ˈʃʊdənt and ˈitən, but it might just be a tiny little schwa-like sound.
And then, of course, native speakers fill in missing words from context. If I carefully said "I shouldn't eaten it," many native speakers might believe they had HEARD "I shoudn't've eaten it."