I have been tasked to secure this data. Either encrypt these two fields or save them in the storage of the phone.
Modern smartphones (Android, iOS) already encrypt their flash when a pin or passcode is set. Your app's storage should be isolated and cannot usually be read by other apps. Assuming your app doesn't ask your user for a master password, any additional encryption you add can just be defeated by reverse-engineering your app. If you could store a per-user encryption key on the server side then you could just store the name and pass on the server anyway, so that's a pointless approach.
How risky is to have the first name and last name saved in clear. Is it a security risk?
I would call it a minimal security risk in most contexts except a few. If your app contains medical data or allows medical data to be inferred (e.g. if your app was for, say, a HIV support group) then you may be beholden to legislation such as HIPAA, not to mention a moral obligation to protect those names. The same goes for things like dating apps where sexual orientation or gender expression are stored, which may be problematic to some people's family situation, or worse under certain governments. You can continue this line of thinking for other threat models of this ilk; effectively, if the use of your app may indicate behaviour or beliefs that could lead to persecution or harm, then you should strongly consider this while developing a security program around your app.
In some jurisdictions a name alone is considered to be personally identifying information (PII) that you have a duty to protect. In the UK, for example, this is covered by the Data Protection Act (DPA). However, the general way this legislation is written usually only makes you liable if you were negligent. I don't think that storing a name alone, in an app's isolated storage, could be considered negligent. The key factor here is that the PII is stored using standard practices (app isolated storage) on a device that the user controls, and the user is responsible for the security of their own device. If they root their device and install a shady app that steals their data, you cannot control for that.