Here's the corrected implementation of the `JobQueue` class with proper synchronization to eliminate the race condition:

```python
import threading

class JobQueue:
    def __init__(self):
        self.jobs = {}
        self.results = {}
        self.lock = threading.Lock()

    def add_job(self, job_id, data):
        with self.lock:
            self.jobs[job_id] = data
            return job_id

    def process_job(self, job_id, processor):
        with self.lock:
            if job_id not in self.jobs:
                return False
            data = self.jobs[job_id]
            del self.jobs[job_id]  # Claim the job to prevent reprocessing

        # Process the job outside the lock to avoid blocking other threads
        result = processor(data)

        with self.lock:
            self.results[job_id] = result

        return True

    def get_result(self, job_id):
        with self.lock:
            return self.results.get(job_id)
```

### Key Fixes and Rationale

1. **Thread Safety**: A `threading.Lock()` is introduced and used to protect all access to shared state (`self.jobs`, `self.results`).

2. **Atomic Job Claiming**: When a job is processed, it is deleted from `self.jobs` within the critical section. This ensures that no other thread can start processing the same job.

3. **Non-blocking Processing**: The actual job processing (`processor(data)`) is performed outside the lock to allow other threads to continue working on different jobs.

4. **Consistent Result Writing**: The result is written back to `self.results` under the lock to ensure consistency and avoid race conditions during concurrent result updates.

5. **Safe Result Retrieval**: The `get_result` method acquires the lock when reading from `self.results`, ensuring thread-safe access.

### Example Usage

```python
import time

def simple_processor(data):
    time.sleep(0.1)  # Simulate work
    return data * 2

queue = JobQueue()
queue.add_job(1, 5)
queue.add_job(2, 10)

print(queue.process_job(1, simple_processor))  # True
print(queue.get_result(1))                     # 10
```

This implementation ensures correct concurrent behavior, prevents data races, and maintains job integrity across threads.