I did find a source for this saying though, it's a children's prayer: http://sv.wikisource.org/wiki/Gud_som_haver_barnen_k%C3%A4r
I found a site that offers several different translations of this prayer here: http://finlander.genealogia.fi/sfhswiki/index.php/Musings_on_%E2%80%9CGud_som_haver%E2%80%9D
'Easy come, easy go' pretty much has the same meaning in English: sometimes life is easy (happy), sometimes not so much. Some might argue that the English expression is more about wealth or success, but in English-speaking cultures wealth and happiness are unfortunately still pretty much synonyms... ;)
After quick search on the internet it doesn’t appear to have the same meaning at all. https://dictionary.cambridge.org/dictionary/english/easy-come-easy-go “Easy” does not mean “happiness”. According to the dictionary the phrase appears to mean that if you didn’t work hard or took risk to obtain something for what others would have to, you will likely not value it appropriately and therefore your chances of loosing it will be generally higher.
An important distinction between the words happy and joyful/glad might need to be pointed out. Happy (Lyckan) is indeed based upon hap, or luck. That is, circumstance or luck dictates happiness. On the other hand, joy or gladness is not necessarily conditional. Many people go through horrendous events and remain at peace in their mind. Some say I split hairs, but so be it.