You should have defined the variables email and password outside the method registerUser(). It seems that your Button is not correctly set as well.

package com.safariagaming.flix;

/**
 * Imports...
 */

public class SignUpScreen extends AppCompatActivity implements View.OnClickListener {

   private String email = null;
   private String password = null;
   private FirebaseAuth.AuthStateListener mAuthListener;

   @Override
   protected void onCreate(Bundle savedInstanceState) {
       super.onCreate(savedInstanceState);
       setContentView(R.layout.activity_sign_up_screen);

  /**
   * Inits and code...
   */

    // Signup button on listener
    buttonSignup.setOnClickListener(new View.OnClickListener() {
    @Override
    public void onClick(View view) {
        email = editTextEmail.getText().toString().trim();
        password = editTextPassword.getText().toString().trim();

        // Verifications
        if (TextUtils.isEmpty(email)) {
            Toast.makeText(LoginActivity.this, R.string.empty_email, Toast.LENGTH_SHORT).show();
            return;
        }

        if (!Patterns.EMAIL_ADDRESS.matcher(email).matches()) {
            Toast.makeText(LoginActivity.this, R.string.wrong_email, Toast.LENGTH_SHORT).show();
            return;
        }

        if (TextUtils.isEmpty(password)) {
            Toast.makeText(LoginActivity.this, R.string.empty_password, Toast.LENGTH_SHORT).show();
        }

        // Authenticating user
        mAuth.createUserWithEmailAndPassword(email, password)
                .addOnCompleteListener(LoginActivity.this, new OnCompleteListener<AuthResult>() {
                    @Override
                    public void onComplete(@NonNull Task<AuthResult> task) {


                        if (!task.isSuccessful()) {
                            Toast.makeText(LoginActivity.this, R.string.wrong_credentials, Toast.LENGTH_SHORT).show();
                        } else {
                            goDashboard();
                            finish();
                        }
                    }
                });
             }
         });
      }
   // onStart() state listener
   @Override
    public void onStart() {
        super.onStart();
        mAuth.addAuthStateListener(mAuthListener);
    }

    @Override
    public void onStop() {
        super.onStop();
        if (mAuthListener != null) {
             mAuth.removeAuthStateListener(mAuthListener);
        }
    }
 }