From what I've seen, 글자 seems to just refer to "character blocks" in general (whether they be Korean syllable blocks or Chinese characters). For example, where we would say a word is "5 letters" in length they would tend to use something like "2 글자" instead referring to the number of blocks, not the count of the letters inside.
However, it sounds like you are referring to the "block shape" itself into which 한글 and 漢字 are written. If that's the case, then I'm not really sure. However...I have seen 네모 used to refer to the "blanks" that are filled in on a form, if that helps any. For example, this is from my SRS deck:
글을 읽고 네모 안에 답을 쓰시오.
Read the text and write your answer in the box.
When I looked 글자 up on Naver's dictionary a lot of the translations seem to be "letter." I need to contrast the syllabic block with 자모, so I'm not sure if 글자 is potentially ambiguous or if the translations on Naver just aren't particularly accurate.
네모 seems like it could work if I just phrase what I'm saying a bit differently. I'll probably go with that unless a native speaker has any input on a specific term for square-shaped syllabic blocks. Thanks!
I know the examples you mean (as I double-checked 글자 with Naver when making my post), but I suspect that is more due to the fact that the equivalent concepts in English would often use "letters" instead. FWIW, 자모 is the word I've usually seen used with actual 한글 letters (ㄱ,ㄴ,ㄷ, etc.) in the past.
I hadn't actually even heard of 음절 until this post, but it wouldn't surprise me if it were referring more to the spoken syllable instead since the first character in 漢字 form is 音 which means "sound" (and many of the Naver examples are about syllable stress).
Regardless, I know there are several native Korean speakers that help out here, so hopefully one of them will chime in soon.
Thanks kenjoluma, that clarifies a lot on the terminology. By "syllabic blocks" I meant the square shape of each syllable. I guess it's something more notable or unusual for people that don't speak Korean. In Korean it doesn't seem like it's remarkable enough for there to be a unique term describing the shape of the syllables, so I'll just use 네모.