Vejo que sua consulta está correta.
Mas se você deseja selecionar com base no ID do carro, deve adicionar o ID do carro como condição também à sua consulta
Por exemplo
SELECT b.id, b.startDate, b.endDate, b.car.id
FROM BorrowedDate b
WHERE :userDateStart NOT BETWEEN b.startDate AND b.endDate
AND :userDateEnd NOT BETWEEN b.startDate AND b.endDate
AND b.car.id = :carId
OU você pode fazer assim também
SELECT b.id, b.startDate, b.endDate, c.id
FROM BorrowedDate b INNER JOIN b.car c
WHERE :userDateStart NOT BETWEEN b.startDate AND b.endDate
AND :userDateEnd NOT BETWEEN b.startDate AND b.endDate
AND c.id = :carId
então você tem que passar :carId value é igual a 2
ATUALIZAÇÃO
você pode tentar esta consulta SQL se precisar obter resultados dinâmicos, atualizarei em breve com JPQL
SELECT b.id, b.startDate, b.endDate, b.car.id
FROM BorrowedDate b
WHERE '2017-04-02 00:00:00' NOT BETWEEN b.startDate AND b.endDate
AND '2017-04-10 00:00:00' NOT BETWEEN b.startDate AND b.endDate
AND b.car.id NOT IN (SELECT DISTINCT bd.car.id FROM BorrowedDate bd WHERE '2017-04-02 00:00:00' BETWEEN bd.startDate AND bd.endDate OR '2017-04-10 00:00:00' BETWEEN bd.startDate AND bd.endDate)
AQUI estão JPQL também
SELECT model.id, model.startDate, model.endDate, model.car.id
FROM BorrowedDate model
WHERE :userDateStart NOT BETWEEN model.startDate AND model.endDate
AND :userDateEnd NOT BETWEEN model.startDate AND model.endDate
AND model.car.id NOT IN (SELECT DISTINCT b.car.id FROM BorrowedDate b WHERE :userDateStart BETWEEN b.startDate AND b.endDate OR :userDateEnd BETWEEN b.startDate AND b.endDate)
