"Евгений" is equal to "Eugene" in English, but it's not such a popular name compared to names in Russia. Short forms are "Gene" and "Eug" (Юдж). I wouldn't make a connection between "Eugene" and "Jack".

Jack is a nickname for the name John, which is a very common name for men. It comes from the Latin name Ioannes, and is the English equivalent of Ivan / Иван. Jack could be something like Vanya / Ваня. Jane is a name for women. It's originally derived from a Biblical character called Joanna, and so the Russian equivalent is Ioanna / Иоанна. Женя is from Евгения, which is Eugenia / Eugenie in English. I suppose a nickname might be something like 'Genie' or 'Nia'.

John is a male name, and Jack has historically been a nickname for people called John. However nowadays Jack is a very popular name in its own right. Throughout the 90s and 00s it was consistently in the top 5 names for baby boys in the UK. Jane is a female name. Both John and Jane probably had the same original source, although they are used for different sexes.