I think it is a good idea to add some historical details about the reason why a "character" is also a "syllable block." In short, Hangul was designed to assign one and the only one character for one syllable, by using a combinatoric method to compose each letter.
King Sejong, the original author of the first manual of the writing system (훈민정음 was not only a guide for ordinary people but also an academic paper), observed a few ways that each syllable is sounded. First, there is only one heading consonant, which can be "stressed." It is followed by a vowel and an ending consonant. The vowel is very complex and there's a lot. The ending consonant can have a "silent" part which is not sounded except for certain conditions. Realizing that Korean is such a unique language, he decided to reinvent a whole system of writing, not simply borrowing those of Chinese, Japanese, Tamil, etc.
The result was a surprisingly simple, predictable and math-friendly form of the character. He took advantage of the combinatory theory to compose each character (Source: speculation). That is, there's one and the only one character for one syllable. That said, there are "technically possible" characters that are impossible to recreate due to the inconsistencies among the consonants. ("쟈" used to be possible when "ㅈ" was simply a voice version of "ㅅ", so /z/. It's changed to /dj/ so adding a /ja/ sound is just as same as adding an /a/ sound.)
When the King was alive, he spoke Middle Korean. Middle Korean has tones, but that was not included in the initial design. Instead, small dots were placed adjacent to a letter to indicate its tone. Oh, but I forgot their name...
So going back to your question! What is a "syllable block" called in Korean? A "character," because there is no need to call them differently. They've got a custom-created set of characters.
Hey, you are wrong. There are syllables that cannot be expressed with a Hangul character.
You're right. There is, now. One example I've learned in school is "사겨" (from "사귀다"). Formally it's "사귀어" but we actually say it like "사겨" (and we write "사겨" online). However, it's just close to "사겨"'s sound, not the same. It should be "사귀ㅓ", but this combination simply doesn't exist.