Hieroglyphic writing systems generally were all initially used to record the accomplishments of kings in a very simple way. The problem they have is that true literature requires using a lot of the language, and logographic systems are really only extensible by adding a new symbols, or tacking other types of systems onto them. To tell an actual story with any kind of evocative language requires development of thousands of unique glyphs, or something more akin to an alphabet.
Egypt followed this pattern. Slowly their writing system started to adopt more advanced concepts (such as phonograms and determinatives), but that didn't happen until around the time of the writing of the Epic of Gilgamesh, so they can't really be said to predate it.
For example, what is considered the oldest known sentence in Egyptian is just the 5 symbols for "unite", "upper and lower Egypt", "son", "king of upper and lower Egypt", and a particular king's symbol, along with a few assorted determinatives written on them. That translates out to roughly "He united upper and lower Egypt for his son, King of Upper and Lower Egypt Perisben". This was probably about 500 years before the Epic of Gilgamesh, depending on how you mesh the two chronologies.
By the time of Middle Egyptian they had expanded the system to include about 900 different hieroglyphs, at which point they could begin to tell proper stories. However, this (barely) post-dates tEoG. Most of the more famous examples of Hieroglyphs on monuments are from this period. There was also a related written form (initially called Hieratic) which slowly over the centuries evolved into a proper alphabet*. Coptic is a living descendent of this system.
* - Abjad technically, since Semitic languages like Egyptian had no real need for vowel glyphs.