Mylroie/Mylroi/Mylrea/Mulray/Milray/Millray and even Mybrea... can be a bit tricky!
The earliest Mylroie family I've come across is one from Ballacosney/Ballachosney in Lonan. John born c1824, in Lonan. He was 37 years old and head of the family in the 1861 census. He was a miner. Which is probably what your Mr Mylroie was doing in Egremont.
There are some earlier Mylroi's [guess this is just a slight variant in the spelling], also in Lonan, at a place called Grane [?]. Judith and Thomas, both aged 55 in the 1841 census. Eldest son is John, age shown as 25 (25 to 29 age range?), born c1816 to c1820. I don't know if this John and the one above are one and the same... they may not be - there is a difference in ages of between 4 and 9 years. The next son was James aged 22. Then Elizabeth, 19.
Don't know if that helps!