How to grant SQL Agent proxy to a public role in MSSQL 2014
Before I answer your question I want to draw your attention about granting permissions to Public Msdb Role on the Proxy.

Reference: Best Practices in Configuring SQL Server Agent Proxy account.


  Proper care must be taken when granting permissions to a security
  principal on a Proxy especially when granting permissions to Public
  Msdb Role on the Proxy. Every user in the database is a member of the
  public role and as a result every user gains access to the Proxy when
  Public Msdb role is granted access to the Proxy. You can query the
  sysproxylogin system table to get the list of SQL Server logins that
  are associated with each SQL Server Agent proxy account.


You can achieve this by using the following TSQL (assuming your proxy is created properly).

USE msdb ;  
GO  

EXEC dbo.sp_grant_login_to_proxy  
    @msdb_role = N'public',  
    @proxy_name = N'Proxy Name' ;  
GO 


Details about using sp_grant_login_to_proxy.