Short answer: no.
Long answer: First off, WhatsApp doesn't have a backdoor so much as it has a minor vulnerability in its end-to-end encryption scheme.
Here's how it works: when you send a message on WhatsApp, it is encrypted before it leaves your phone, sent through the internet, and never decrypted until it lands on the recipient’s phone. Only the recipient’s phone contains the key that can decrypt the message. Private keys are generated and exchanged between users before any messages are ever sent.
But if the recipient changes their phone, they no longer have the encryption key necessary to decrypt messages. This can result in messages that are never delivered. In a compromise of security in favor of convenience, WhatsApp implemented what many critics allege is a backdoor. The key exchange is renegotiated without the recipient's knowledge, and the backlog of messages are re-sent. WhatsApp users can toggle a preference in their settings to let them know if the recipient has changed phones and the keys have been renegotiated, but this is not enabled by default.
The result is that if someone simply turns off their phone or is disconnected from the internet for any period of time, WhatsApp--and ergo Facebook--could fake the existence of a new phone and private key in order to read someone's message history.
To do this, one would need WhatsApps explicit compliance and a specific target. This is not something that can be exploited through any sort of mass dragnet surveillance or by hackers without WhatsApp's help. So for most people, it is not a concern. If you feel you are being targeted to such a degree, you can find more details on this issue and read up on available WhatsApp alternatives.
Back to our original question, a VPN will not protect you in any way from this vulnerability. The exploit would have to occur on WhatsApp's servers, not on your internet connection or device.