Hi Paan,
You don't NEED to change your DNS server (NS1 and NS2).
This is not the problem.
The problem is, that you are trying to receive mail for the same domain, in to different places.
To clarify.
When a mail-server sends a mail to @peroneus.com, the mail-server asks the DNS-server where to send the mail.
This is called a MX record.
A MX record tells mail servers. "Send mail to this domain, to the server with this name"
The server name is a A-record.
For your domain:
peroneus.com MX preference = 10, mail exchanger = mailstore1.secureserver.net
peroneus.com MX preference = 0, mail exchanger = smtp.secureserver.net
This tells us, that our mail-server will try to send mail to two servers. If the first one does not answer, the next one will be contacted.
So in order for mails to be send to Office365, those information has to be changed to something like:
peroneus.comt MX preference = 20, mail exchanger = peroneus.com-mail.eo.outlook.com
However, if you do this, mails will no longer be send to your initial servers, resulting in none of the other employees will receive mail.