To build a minimal linear regression model using Flux.jl in Julia to test if an integer is even, we need to define a set of input and output data, and then train the model using these data. However, linear regression is not an appropriate method for testing if an integer is even, as this is a classification problem and not a regression problem. Instead, we can use a simple function to check if an integer is even or odd.

Here's an example code:

```julia
using Flux

# Define a function to check if an integer is even or odd
function iseven(x)
    if x % 2 == 0
        return 1
    else
        return 0
    end
end

# Define the input and output data
inputs = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
outputs = [iseven(x) for x in inputs]

# Define the model
model = Chain(Dense(1, 1))

# Define the loss function
loss(x, y) = Flux.Losses.mse(model(x), y)

# Define the optimizer
opt = ADAM()

# Train the model
Flux.train!(loss, Flux.params(model), [(inputs[i], outputs[i]) for i in 1:length(inputs)], opt)

# Test the model
x = 7
prediction = round(model(x)[1])
println("Prediction for $x is $prediction, and it should be $(iseven(x)).")
```
In this example, we define a function called iseven that returns 1 if the input integer is even and 0 if it's odd. We then define the input and output data for our model, where the input data is a set of integers from 1 to 10, and the output data is the result of applying the iseven function to each input integer.

Next, we define a simple model with one input and one output neuron using the Dense function. We also define the mean squared error loss function, which is commonly used for regression problems. We then define the Adam optimizer, which is a popular optimizer for neural networks.

Finally, we train the model using the train! function from Flux. Once the model is trained, we can test it by passing an integer to the model and comparing the output to the result of the iseven function for that integer.
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