It is not a complusion to wash right part first and left afterward during Ghusl. For a Ghusl to be valid, one needs to wash their bpdy thoroughly ( i.e. that shower needs to be a proper shower, with due effort and due regard given to achieving proper cleanliness). Once that has been done, there is no need for separate wudu.
Regarding what Quran says how Ghusl should be done, please see:
Discussion 30957 • Reply 30977
Whether Wuzu is compulsory for Ghusl, please see:
Discussion 13304 • Reply 37005
However, it is recommended that one follows uswa of the Prophet (sws) while performing Ghusl as well (as described in the above link).
Also Prophet (sws) always preferred right hand over left while dealing with day to day affairs of life. Therefore, if one wants to adopt this sequence while doing Ghusl, then it is a commendable act too. But it is not a religious injunction.
Please see for details:
Discussion 56199