Add user authentication and game persistence with SQLite

- User registration/login with bcrypt password hashing
- SQLite database with goose migrations and sqlc-generated queries
- Games and players persisted to database, resumable after restart
- Guest play still supported alongside authenticated users
- Auth UI components (login/register forms, auth header, guest banner)

game/types.go

@@ -1,9 +1,12 @@
 package game
 
+import "encoding/json"
+
 type PlayerID string
 
 type Player struct {
 	ID       PlayerID
+	UserID   *string // UUID for authenticated users, nil for guests
 	Nickname string
 	Color    int // 1 = Red, 2 = Yellow
 }
@@ -35,3 +38,27 @@ func NewGame(id string) *Game {
 		Status:      StatusWaitingForPlayer,
 	}
 }
+
+func (g *Game) BoardToJSON() string {
+	data, _ := json.Marshal(g.Board)
+	return string(data)
+}
+
+func (g *Game) BoardFromJSON(data string) error {
+	return json.Unmarshal([]byte(data), &g.Board)
+}
+
+func (g *Game) WinningCellsToJSON() string {
+	if g.WinningCells == nil {
+		return ""
+	}
+	data, _ := json.Marshal(g.WinningCells)
+	return string(data)
+}
+
+func (g *Game) WinningCellsFromJSON(data string) error {
+	if data == "" {
+		return nil
+	}
+	return json.Unmarshal([]byte(data), &g.WinningCells)
+}