That's no good.
To make sure I understand the issue you are experiencing - are you saying that you get SMS when the phone is turned on, but if you put it in airplane mode or turn off, any SMS sent in the time your device was offline do not come through when you turn off flight mode or turn your device back on?
What did Boost advise specifically when you contacted them?
It could be that there is a fault with the SIM card or the actual phone itself which is causing the problem.
Do you have spare phone you can put the SIM card in to see if the problem persists in a different device? I would try this first to and if the problem persists whilst in a different phone, you may need to get a new SIM card. You can get a new SIM and keep the same numbers.
You can go online and purchase a SIM card directly from the Boost Website Shop or you can also find your closest Boost Retailer to purchase a SIM.
If you still have trouble there might be a problem with you phone itself in which case you can take it to be repaired.
Otherwise I would recommend that you chat to Boost Support again on 125 8881 (8am-9pm, 7 days), advise them that the issue is still occurring to see what they can do further to assist.
Let us know how you go.