In order to get Mime Jr. from an egg, you need to either breed a Mr. Mime or Mr. Rime while they're holding and odd incense, which can be brought in Hulbury. There are baby Pokemon that will only hatch from an egg if their parent holds a certain incense. Here is a list of baby Pokemon that hatch when the parent Pokemon hold an incense and which one they will need if you also wanted other baby Pokemon in the future.
If you also want to catch a wild Mime Jr. you can find them in the following locations:
Slippery Slope, Frostpoint field, Giant's Bed, Old Cemetery, Snowslide Slope, Giant's Foot, Frigid Sea, Three-Point Pass and Ballimere Lake
Source
Hope this helps!
