I understand that 好 in 睡不好觉 adds the meaning of "well", so "sleep well" instead of "sleep", but does 下 in 吃不下饭 change anything?
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Yes it does.
I won't go into the details of your textbook example. It may not be the best one because in that sentence the 下 could probably be replaced by some other result complement and roughly keep the same translation in English.
Generally speaking, 下 as a result complement often has the meaning of fitting or containing (the potential complement as in "V 得 Res." and "V 不 Res." constructions is similar to the result complement).
他车子能坐下5个人 (= his car can accommodate 5 people)
电梯里站得下10个人 (= ten people can fit in the elevator = the elevator can contain ten people)
这个箱子太小了，装不下这么多衣服 (= this box is too small, it can't contain so many clothes)
我吃不下饭 (= I can't eat any more = I can't fit any more food in my stomach, or, my stomach can't contain any more food)
As additional proof of the semantic value of 下 in this construction, you can replace it with another result complement to a different effect:
我吃不了饭 (= I can't eat = meaning that something is preventing you from doing so, instead of your stomach being unable to eat more, e.g. 今天我工作太忙，吃不了饭)
吃不下饭 can be found in Adsotrans as:
to be unable to eat
吃不下 Is recorded in multiple dictionaries. Here’s the entry from Pleco C-E:
not feel like eating; be unable to eat any more (opp. 吃得下)
Tā bù tài shūfu, chī bu xià fàn.
He’s not very well and doesn’t feel like eating.
Xièxie, wǒ shízài chī bu xià le.
Thanks, but I really can’t eat any more. or Thanks, I’ve really had enough.
The reason I included this is because it includes the opposite: 吃得下.
Some verbs can be qualified with 得下 or 不下 indicating ability or lack thereof.